Endangered Whale Populations through Mathematical Modeling
The Past, Present, and Future of
Endangered Whale Populations
:
Qualitative Analysis and
Modeling
1.
Mathematical Modeling
1.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
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Key Question:
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?
1.
Mathematical Modeling
2.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
Resource Management
•
Why have natural resources, such as whales or
bison, been depleted so quickly?
•
How can we restore natural resources?
•
How should we manage natural resources?
1.
Mathematical Modeling
2.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
General Biological Resource Model
Let
X
be the biomass of resources.
Let
T
be the time.
Let
G
(
X
)
be the resource growth rate.
Let
H
(
X
)
be the consumption per consumer.
Let
C
be the (
fixed
) number of consumers.
Overall rate of increase = growth rate – consumption rate
1.
Mathematical Modeling
1.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
Growth Model
Overall rate of increase = growth rate – consumption rate
•
Assume logistic growth
•
Holling type 3 model
–
Saturation and alternative resource
Consumption Model
The Full Model
Overall rate of increase = growth rate – consumption rate
1.
Mathematical Modeling
2.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
Nondimensionalization
Overall rate of increase = growth rate – consumption rate
This model has 5 parameters—a lot for analysis!
Nondimensionalization reduces the number of parameters.
1.
R
e
p
l
a
c
e
X
b
y
x
=
X
/
K
a
n
d
T
b
y
t
=
R
T
.
2.
Define new dimensionless parameters.
Dimensionless Version
p
represents the processing-discovery ratio.
c
represents the number of consumers.
1.
Mathematical Modeling
2.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
The resource increases
The resource decreases
Q
u
a
l
i
t
a
t
i
v
e
A
n
a
l
y
s
i
s
represents growth
represents consumption
E
x
a
m
p
l
e
:
p
=
0
.
0
1
E
x
a
m
p
l
e
:
p
=
0
.
0
1
,
c
=
0
.
1
2
Low consumption – high resource level
Curve
above line
Growth > Consumption
Population increases
E
x
a
m
p
l
e
:
p
=
0
.
0
1
,
c
=
0
.
3
6
High consumption – low resource level
Curve
below line
Growth < Consumption
Population decreases
E
x
a
m
p
l
e
:
p
=
0
.
0
1
,
c
=
0
.
2
4
Growth
> Consumption
Population increases
Growth
< Consumption
Population decreases
Moderate consumption – two resource levels
1.
Mathematical Modeling
2.
A Resource Management Model
A.
The general plan for the model
B.
Details of growth and harvesting
C.
Simplification
D.
Analysis of the model
E.
Application to whale populations
W
h
a
l
e
C
o
n
s
e
r
v
a
t
i
o
n
•
Can we use our general resource model for
whale conservation?
•
Issues:
–
Model assumes fixed consumer population.
•
We’ll look at distinct stages.
–
Model assumes harvesting with uniform
technology.
•
Advanced technology can be thought of as more
consumers.
Stage 1 – natural balance
Low consumption
High population
Stage 2 – depletion
Consumption increases
Population decreases
and then crashes
Stage 3 – inadequate correction
Consumption decreases
Population increases,
but only a little
Consumption
sustainable,
but population can’t
recover
Stage 4 – recovery
Consumption decreases
Eventually only a high
population is stable
We need population
above the
unstable equilibrium
before we can increase
consumption.
Stage 5 – proper management
Moderate consumption
is sustainable, provided
population first
recovers beyond the
unstable equilibrium.
Thanks for coming!
gledder@unl.edu
Explore the past, present, and future of endangered whale populations through qualitative analysis and mathematical modeling. Delve into resource management models, input data analysis, species control parameters, and the importance of managing natural resources for the conservation of whales. Learn about the relationship between input and output data, future population dependence on control parameters, and the general biological resource model.
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Presentation Transcript
The Past, Present, and Future of Endangered Whale Populations: Qualitative Analysis and Modeling Glenn Ledder University of Nebraska-Lincoln https://www.math.unl.edu/~gledder1 gledder@unl.edu
1. Mathematical Modeling 1. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
Mathematical Model Input Data Math Problem Output Data Key Question: What is the relationship between input and output data?
Endangered Species Fixed Parameters Mathematical Model Future Population Control Parameters Model Analysis: For a given set of fixed parameters, how does the future population depend on the control parameters?
1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
Resource Management Why have natural resources, such as whales or bison, been depleted so quickly? How can we restore natural resources? How should we manage natural resources?
1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
General Biological Resource Model Let X be the biomass of resources. Let T be the time. Let G(X) be the resource growth rate. Let H(X) be the consumption per consumer. Let C be the (fixed) number of consumers. ?? ??= ?(?) ? ?(?) Overall rate of increase = growth rate consumption rate
1. Mathematical Modeling 1. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
Growth Model Assume logistic growth ?? ??= ?? 1 ? ? ?(?) ? Overall rate of increase = growth rate consumption rate
Consumption Model Holling type 3 model Saturation and alternative resource Q 0.75Q ??2 ?2+ ?2 ?(?) = G 0.5Q 0.25Q 0 0 A 2A X 3A 4A
The Full Model ??2 ?2+ ?2 ?? ??= ?? 1 ? ? ? Overall rate of increase = growth rate consumption rate
1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
Nondimensionalization ??2 ?2+ ?2 ?? ??= ?? 1 ? ? ? Overall rate of increase = growth rate consumption rate This model has 5 parameters a lot for analysis! Nondimensionalization reduces the number of parameters. 1. Replace X by x = X/Kand T by t = RT. 2. Define new dimensionless parameters.
Dimensionless Version ? =? 1 ? ?, ? = ??, ? = ???, ? = ??? 1 ? ?? ??= 1 ? ?? ? + ?2 p represents the processing-discovery ratio. c represents the number of consumers.
1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
Qualitative Analysis ?? ??= ? ? ? ? + ?? ?? ? + ?2 ? ? = (? ?)(? + ??) represents growth ? ? = ?? represents consumption 1 ? ? + ?2> ?? The resource increases 1 ? ? + ?2< ?? The resource decreases
Example: p=0.01 The consumption function is ?? The growth function is (? ?)(? + ??)
Example: p=0.01, c=0.12 Curve above line 1 ? ? + ?2> ?? Growth > Consumption Population increases Low consumption high resource level
Example: p=0.01, c=0.36 Curve below line 1 ? ? + ?2< ?? Growth < Consumption Population decreases High consumption low resource level
Example: p=0.01, c=0.24 If ? ?< ? < ?.??: Growth > Consumption Population increases If ?.?? < ? <? ?: Growth < Consumption Population decreases Moderate consumption two resource levels
1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations
Whale Conservation Can we use our general resource model for whale conservation? Issues: Model assumes fixed consumer population. We ll look at distinct stages. Model assumes harvesting with uniform technology. Advanced technology can be thought of as more consumers.
Stage 1 natural balance Low consumption High population
Stage 2 depletion Consumption increases Population decreases and then crashes
Stage 3 inadequate correction Consumption decreases Population increases, but only a little Consumption sustainable, but population can t recover
Stage 4 recovery Consumption decreases Eventually only a high population is stable We need population above the unstable equilibrium before we can increase consumption.
Stage 5 proper management Moderate consumption is sustainable, provided population first recovers beyond the unstable equilibrium.
Thanks for coming! gledder@unl.edu
